Is Pi in the Sky? Where Do You Stand on the Question?

"It is written that animals are divided into (a) those that belong to the Emperor (b) embalmed ones (c) those that are trained (d) suckling pigs (e) mermaids (f) fabulous ones (g) stray dogs (h) those that are included in this classification (i) those that tremble as if they were mad (j) innumerable ones (k) those drawn with a very fine camel's hair brush (l) others (m) those that have just broken a flower vase (n) those that resemble flies from a distance." ---Jorge Luis Borges,"The Analytical Language of John Wilkins" (in Other Inquisitions)

What is the nature of the vast body of concepts, experiences, perceptions, formulations, practices, computations, applications, that is called mathematics? More particularly, what is the nature of the relationship between the people who are engaged with these things and the things themselves? I've singled out mathematics, but the same questions can be asked of any branch of science.

Most working mathematicians and scientists ignore these questions as irrelevant to the daily pursuit of their respective specialties. But the last question especially---the one I'm concerned with here---is one of the basic questions of the philosophy of science; it has been considered by the finest brains. The question has been around for three thousand years and still has plenty of fuel left in it to arouse passions to the flash point and beyond. And answers do occasionally influence educational policy. One answer is that mathematics exists independently of humans. Its truths are universal, independent of time, and valid in all conceivable worlds-and perhaps even in the absence of worlds. The job of the mathematician is merely to dig out, validate, and display these truths. This is an extreme view, one that has often been parodied in the form of the assertion that "pi is in the sky." This set of beliefs I call platonism.

Diametrically opposed to platonism is the view that mathematics is a social product, the creation or the construction of groups of communicating people, and that the universe could conceivably have gotten along-thank you very much-without a jot of mathematics. This statement is an extreme form of what is called "social constructivism," a philosophy that has been gaining strength and popularity during the present century. Among mathematicians its adherents still constitute a small minority, since most mathematicians, if they think consciously of philosophy at all, will describe themselves as platonists.

Ian Hacking, a professor of philosophy at the University of Toronto, has given us a number of stunning books on the history and philosophy of science (I think particularly of his Emergence of Probability). He now takes up with equal zest the question of social constructivism, relating it to specific areas in the natural sciences, weapons research, mental health, rocks (the early creation of masses of dolomite), child abuse, and (!) the question of whether Captain Cook was considered a god by the natives of Hawaii. An odd mixture, you might say, but a mixture concocted from the dozens of topics---including gender, quarks, and Zulu nationalism---about which books have now been written along social constructivist lines. Search your library holdings for "social constructivism" and you will see.

Given his knowledge of mathematics and its history, I was disappointed that Hacking did not take up the case of mathematics in any detail, dismissing the major work of Paul Ernest, Social Constructivism as a Philosophy of Mathematics (which he cites), with the niggling complaint (page 48) that Ernest might have avoided a certain ambiguity by employing the word "constructionism" in his title instead of the word "constructivism."

Oh well, authors are not required to write the books that either their editors or their reviewers would like to see. Since, reciprocally, reviewers have the freedom to pick and choose in discussing an author's pages, I prefer to pass over the case studies with their specifics, with their conflicts of belief, their endlessly detailed nuances of opinions, nuances that exhibit what Sigmund Freud called the "narcissism of small differences," and proceed directly to Hacking's personal choices of the abstract principles that make up the nub of social constructivism.

These principles have been formulated quite forcefully in what Hacking terms three "sticking points." I suppose he means by his choice of adjective that the major philosophical positions by which he characterizes the constructivist position will stick in the throats of those who happen to believe otherwise. And this would include most scientists.

Sticking Point #1 (page 78). Contingency. The subject could have developed in a totally different way, leading to a nonequivalent corpus of material. In his discussion here, Hacking sets out shades of abstract belief and disbelief, which, I believe, will be confusing to the na´ve outsider.

The opposite of contingency is necessity. But physicists, says Hacking, are not "necessitarians." They are "inevitabilists," believing that if successful physics has taken place, it was inevitable that it take place along the lines that it did.

Sticking Point #2 (page 83). Nominalism. "The world is so autonomous . . . that it does not even have what we call structure in itself. . . . All the structure we can conceive lies within our representations." The anti-nominalists would assert "inherent structurism. . . . I suppose that most scientists believe that the world comes with an inherent structure which is their task to discover."

Sticking Point #3 (page 92). Explanations of Stability of Theories. "The constructionist holds that explanations for the stability of scientific belief involve, at least in part, elements that are external to the professed content of the science. These elements typically include social factors, interests, networks. . . . Opponents hold that whatever be the context of discovery, the explanation of stability is internal to the science itself."

Hacking believes---as I do---that very few people will accept or reject these sticking points outright, and so he allows for shades of belief. He invites his readers to rate their adherence to his three sticking points on a scale of 0 to 5, 0 for total disbelief in social constructivism, 5 for agreement with the positions advocated. (Question: Do rating systems reflect an objective reality, or are they socially constructed?)

How does Hacking rate himself (page 99)? On Contingency: 2. On Nominalism: 4. On Stability: 3. This last rating seems to worry him more than the others, and he discusses it in a long footnote. Where is this vaunted stability? he seems to be asking, noting (page 206) that "The dolomite problem leaves philosophical questions of stability untouched, precisely because it is still a problem."

On the basis of these ratings, I would call Hacking a mild constructivist. Whom does he dub the paradigm of constructivists? The most famous, or certainly the most discussed, philosopher of science of our times, the man who took the word "paradigm" out of the appendix of Latin grammars and put it on the lips of all philosophers and humanists: the late Tom Kuhn.

Hacking rates Kuhn 5, 5, 5, all the way (page 99).

My own self-rating would be: 4, 4, 4, which is that of a fairly strong constructivist.

When I was younger, I believed that the G minor symphony of Mozart was note-perfect, existing in heaven before the world was even created. Later, I saw the struggles recorded on a page of a manuscript score. Thinking of this experience, I might up my rating on contingency to 4.5.

When I remind myself of Borges' parody of the way we make categories (cited in the epigraph to this article), I might up my rating on nominalism to 5.

I now invite readers to rate themselves. But before they do, let them introspect on where they stand on homeopathy, acupuncture (both encouraged by mathematician/philosopher Paul Feyerabend); on IQ testing; on whether measures of missile accuracy are an objective concept (think of the Kosovo experience); on the existence of nanobacteria; on whether the true TRH (thyrotropin-releasing hormone) has been isolated; on whether magnetic monopoles exist (on which question I've heard that a great deal of travel money has been spent). Some of these topics are discussed in Chapter 6.

Mathematical readers may object to these pro-constructivist examples. Mathematics is different, they will say. The contraction mapping theorem, the axioms for a group, for example, have nothing to do with us or with the exterior world. Mathematics enjoys a unique position among the scientific disciplines. Or does it? Answer this: Would there be any mathematics if the universe contained no sentient creatures? And would a positive answer to this question be objective knowledge or socially constructed theology?